Integrating factor with non-standard form

7

1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$ is \(x + \sqrt { x ^ { 2 } - 1 }\).
2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$ for which \(y = 1\) when \(x = \frac { 5 } { 4 }\). Give your answer in the form \(y = f ( x )\).

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \left( 1 + \frac { 3 } { x } \right) = \frac { 1 } { x ^ { 2 } } , \quad x > 0$$

1. Verify that \(x ^ { 3 } \mathrm { e } ^ { x }\) is an integrating factor for the differential equation.
2. Find the general solution of the differential equation.
3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\).
   (3)(Total 10 marks)

7

1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).

7

1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).

4 The differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { 1 - x ^ { 2 } } y = ( 1 - x ) ^ { \frac { 1 } { 2 } } , \quad \text { where } | x | < 1$$ can be solved by the integrating factor method.

1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left( \frac { 1 + x } { 1 - x } \right) ^ { \frac { 1 } { 2 } }\).
2. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = \mathrm { f } ( x )\).

2

1. Show that \(\frac { 1 } { x ^ { 2 } }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = x$$
2. Hence find the general solution of this differential equation, giving your answer in the form \(y = \mathrm { f } ( x )\).

5

1. Show that \(\tan x\) is an integrating factor for the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { \sec ^ { 2 } x } { \tan x } y = \tan x$$ (2 marks)
2. Hence solve this differential equation, given that \(y = 3\) when \(x = \frac { \pi } { 4 }\).
   (6 marks)

3

1. Show that \(\sin x\) is an integrating factor for the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = 2 \cos x$$
2. Solve this differential equation, given that \(y = 2\) when \(x = \frac { \pi } { 2 }\).

5

1. Differentiate \(\ln ( \ln x )\) with respect to \(x\).
2. 1. Show that \(\ln x\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { x \ln x } y = 9 x ^ { 2 } , \quad x > 1$$
   2. Hence find the solution of this differential equation, given that \(y = 4 \mathrm { e } ^ { 3 }\) when \(x = \mathrm { e }\).
      (6 marks)

$$\frac{dy}{dx} + y\left(1 + \frac{3}{x}\right) = \frac{1}{x^2}, \quad x > 0.$$

1. Verify that \(x^3e^x\) is an integrating factor for the differential equation. [3]
2. Find the general solution of the differential equation. [4]
3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\). [3]

The differential equation $$\frac{dy}{dx} + \frac{1}{1 - x^2} y = (1 - x)^{\frac{1}{2}}, \quad \text{where } |x| < 1,$$ can be solved by the integrating factor method.

1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left(\frac{1 + x}{1 - x}\right)^{\frac{1}{2}}\). [2]
2. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = f(x)\). [6]